Why in proximity the other character mesh is disappearing?
The problem is probably NOT caused by clipping. I'm pretty sure the origin of the rig in this animation is from a certain key frame not inside the camera angle anymore. So the camera makes the mesh for the rig invisible because it concludes that the rig (or certain bones in the armature) can't be seen anyways from this point of view. Especially for walk animations, try to animate the rig as an object first (location) before creating the walk animation to make sure, that the origin is with your character every time.

If this didn't help or your project is too complex for this, you can also try to add a lattice modifier to the mesh. I'm pretty sure it will solve the problem :)